Stability in probability of nonlinear stochastic systems with delay.

*(English. Russian original)*Zbl 0843.93086
Math. Notes 57, No. 1, 103-106 (1995); translation from Mat. Zametki 57, No. 1, 142-146 (1995).

This paper considers a stochastic differential equation with a right hand side that has terms that are linear in delayed state, terms where delayed states are multiplied by Wiener processes, and terms that are nonlinear functions of the delayed state with available upper bounds in terms of the powers of the state. The Lyapunov method is used to derive a sufficient condition for the stability in probability of the null solution.

Reviewer: E.Yaz (Fayetteville)

##### MSC:

93E15 | Stochastic stability in control theory |

93C10 | Nonlinear systems in control theory |

34K35 | Control problems for functional-differential equations |

##### Keywords:

functional differential; stochastic differential equation; delay; Lyapunov method; stability
PDF
BibTeX
XML
Cite

\textit{L. E. Shaikhet}, Math. Notes 57, No. 1, 103--106 (1995; Zbl 0843.93086); translation from Mat. Zametki 57, No. 1, 142--146 (1995)

Full Text:
DOI

**OpenURL**

##### References:

[1] | I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Naukova Dumka, Kiev (1968). · Zbl 0169.48702 |

[2] | V. B. Kolmanovskii and V. R. Nosov, Stability and Periodic Ranges of Regularizable Systems with Persistence [in Russian], Nauka, Moscow (1981). |

[3] | V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, London (1986). · Zbl 0593.34070 |

[4] | E. F. Tsar’kov, Random Perturbations of Functional-Differential Equations [in Russian], Zinatne, Riga (1989). |

[5] | L. E. Shaikhet, Probl. Peredachi Inf.,11, No. 4, 70–76 (1975). |

[6] | L. E. Shaikhet, Povedenie Sistem Sluch. Credakh. Kiev (1975), pp. 52–60. |

[7] | L. E. Shaikhet, Poved. Sistem. Sluch. Cred. Kiev (1976), pp. 48–55. |

[8] | R. Z. Khas’minskii, Stability of Systems of Differential Equations under Random Perturbations of Their Parameters [in Russian], Nauka, Moscow (1969). |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.